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Latent splitting as a causal probe

Santiago Zamora, Pedro Lauand, Isadora Veeren, Davide Poderini, Rafael Chaves

TL;DR

This work addresses the challenge of detecting nonclassicality in quantum networks when observational data and standard Pearl-like interventions on observed variables are insufficient, particularly in space-like separated configurations like the triangle. It introduces latent splitting, a causal-intervention on latent quantum edges, and couples it with the inflation technique to derive robust, analytic witnesses of nonclassicality. The authors prove that Pearl-like do-interventions can be reconstructed from latent-splitting data, and they demonstrate substantial gains in certifying nonclassicality: (i) extending the RGB4 family’s nonclassical region with robust polynomial witnesses, and (ii) certifying nonclassicality in the minimal triangle via a nonlinear Bell inequality, with a Fritz-trick-inspired interventional version showing violations under realistic noise. Overall, latent splitting provides a principled bridge between causal inference and quantum operational theory, strengthening device-independent certification in networks and suggesting avenues for more resilient quantum information protocols in complex topologies.

Abstract

Generalizations of Bell's framework to causal networks have yielded new foundational insights and applications, including the use of interventions to enhance the detection of nonclassicality in scenarios with communication. Such interventions, however, become uninformative when all observable variables are space-like separated. To address this limitation, we introduce the latent splitting procedure, a generalization of interventions to quantum networks in which controlled manipulations are applied to latent quantum systems. We show that latent splitting enables the detection of nonclassicality by combining observational and interventional data even when conventional interventions fail. Focusing on the triangle network, we derive new analytical witnesses that robustly certify nonclassicality, including nonlinear inequalities for minimal binary-variable scenarios and extensions of the nonclassical region of previously proposed experiments.

Latent splitting as a causal probe

TL;DR

This work addresses the challenge of detecting nonclassicality in quantum networks when observational data and standard Pearl-like interventions on observed variables are insufficient, particularly in space-like separated configurations like the triangle. It introduces latent splitting, a causal-intervention on latent quantum edges, and couples it with the inflation technique to derive robust, analytic witnesses of nonclassicality. The authors prove that Pearl-like do-interventions can be reconstructed from latent-splitting data, and they demonstrate substantial gains in certifying nonclassicality: (i) extending the RGB4 family’s nonclassical region with robust polynomial witnesses, and (ii) certifying nonclassicality in the minimal triangle via a nonlinear Bell inequality, with a Fritz-trick-inspired interventional version showing violations under realistic noise. Overall, latent splitting provides a principled bridge between causal inference and quantum operational theory, strengthening device-independent certification in networks and suggesting avenues for more resilient quantum information protocols in complex topologies.

Abstract

Generalizations of Bell's framework to causal networks have yielded new foundational insights and applications, including the use of interventions to enhance the detection of nonclassicality in scenarios with communication. Such interventions, however, become uninformative when all observable variables are space-like separated. To address this limitation, we introduce the latent splitting procedure, a generalization of interventions to quantum networks in which controlled manipulations are applied to latent quantum systems. We show that latent splitting enables the detection of nonclassicality by combining observational and interventional data even when conventional interventions fail. Focusing on the triangle network, we derive new analytical witnesses that robustly certify nonclassicality, including nonlinear inequalities for minimal binary-variable scenarios and extensions of the nonclassical region of previously proposed experiments.
Paper Structure (16 sections, 1 theorem, 49 equations, 7 figures)

This paper contains 16 sections, 1 theorem, 49 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Causal modeling
  4. The triangle scenario
  5. Interventions
  6. Latent Splitting
  7. Results
  8. Pearl interventions from latent splitting
  9. Quantum nonclassicality with latent splitting
  10. Robust Certification and Witnesses
  11. Beyond quantum nonclassicality with latent splitting
  12. Final remarks
  13. Acknowledgments
  14. Pearl interventions and Proof of Theorem \ref{['thm:Pearl_from_partial']}
  15. Equivalencies between latent splitting and Pearl interventions
  16. ...and 1 more sections

Key Result

Theorem 1

Let us consider a two-layered network, with sources $\Lambda_1,\dots,\Lambda_m$, each distributing a system $\rho_i$, and parties $A_1,\dots,A_n$, performing measurements $E_{a_j|Pa^O(a_j)}$. Then any do-conditionals on this network, representing a Pearl-like intervention, can always be inferred by

Figures (7)

  • Figure 1: DAGs describing Bell and triangle scenarios. Roman letters denote observed variables, while greek letters denote latent variables. (a) In a bipartite Bell scenario, a latent variable $\lambda$ influences the outcomes of two parties, with inputs $X,Y$ and outputs $A,B$. Notice that the inputs are just a particular instance of observed variables. (b) In the triangle scenario, three independent sources, given by latent variables $\alpha, \beta, \gamma$, influence the observed variables $A,B,C$.
  • Figure 2: Illustration of a latent splitting. On top (a) is the original causal structure, where party $A_l$ is influenced by several sources. The link from source $\Lambda_k$ to $A_l$ are highlighted in red. In the next panel (b) it is shown the effective causal structure after the intervention, where party $A_l$ has replaced its incoming state from source $\Lambda_k$, effectively severing that causal link.
  • Figure 3: Latent splitting in the triangle scenario. (a) Shows the latent splitting performed by $A$, cutting effectively the influence from $\gamma$. (b) The joint DAG resulting from the intervention.
  • Figure 4: The RGB4 distribution for distinct values of $u$. Here we compare different intervals of certification corresponding to nonclassical correlations $P^u_{obs}$ and $P^u_{int}$. The green lower intervals, $u\in(0,0.223)$ and $u\in(0.816,1)$ indicate the result proved in renou2019genuine, which exclude imperfect preparations or noisy measurements. Our work (red upper intervals) identifies first a broader region, $u\in(0,0.4)$ and, a slightly smaller region, $u\in(0.84,1)$, obtained through the inflation technique.
  • Figure 5: Inflation of the Joint DAG in the triangle scenario. An inflation of the Joint Causal DAG shown in Fig. \ref{['fig: triangle-PI']}(b). The duplicated source is $\beta$, and since it is a classical latent variable it can distribute copies of share randomness to copies of $A$, $\hat{A}$ and $C$.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Definition 1: Latent splitting
  • Theorem 1
  • Definition 2: Pearl intervention